what transformations produce the graph of g(x)=5^-x+2 from the graph of the parent function f(x)=5^x? Select all that apply
reflection over the x-axis
reflection over the y-axis
horizontal shift to the left 2 units
horizontal shift to the right 2 units
vertical shift up 2 units
vertical shift down 2 units
The transformations that produce the graph of g(x)=5^-x+2 from the graph of f(x)=5^x are:
1. Reflection over the x-axis
2. Horizontal shift to the right 2 units
3. Vertical shift up 2 units
To transform the graph of the parent function f(x) = 5^x to the graph of g(x) = 5^(-x+2), we need to apply the following transformations:
1. Reflection over the x-axis: This transformation would result in a reflection of the graph of f(x) about the x-axis.
2. Reflection over the y-axis: This transformation would result in a reflection of the graph of f(x) about the y-axis.
3. Horizontal shift to the left 2 units: This transformation would shift the graph of f(x) to the left by 2 units.
4. Horizontal shift to the right 2 units: This transformation would shift the graph of f(x) to the right by 2 units.
5. Vertical shift up 2 units: This transformation would shift the graph of f(x) upward by 2 units.
6. Vertical shift down 2 units: This transformation would shift the graph of f(x) downward by 2 units.
Therefore, the correct transformations for the graph of g(x) = 5^(-x+2) from the parent function f(x) = 5^x are:
- Reflection over the x-axis
- Horizontal shift to the left 2 units
- Vertical shift up 2 units
To determine the transformations that produce the graph of g(x) = 5^(-x+2) from the parent function f(x) = 5^x, we need to analyze the changes applied to the function.
1. Reflection over the x-axis:
To reflect a function over the x-axis, we multiply the entire function by -1. In this case, g(x) = -5^(-x+2). Therefore, there is a reflection over the x-axis, and this transformation applies.
2. Reflection over the y-axis:
To reflect a function over the y-axis, we replace x with -x. However, in this case, we replace -x+2 into f instead of x. Therefore, there is no reflection over the y-axis, and this transformation does not apply.
3. Horizontal shift to the left 2 units:
To shift a function horizontally to the left, we subtract the desired amount from x. In this case, g(x) = 5^(-x+2), which means we subtract 2 units from x. Therefore, there is a horizontal shift to the left 2 units, and this transformation applies.
4. Horizontal shift to the right 2 units:
To shift a function horizontally to the right, we add the desired amount to x. In this case, g(x) = 5^(-x+2), which means we add 2 units to x. Therefore, there is a horizontal shift to the right 2 units, and this transformation applies.
5. Vertical shift up 2 units:
To shift a function vertically up, we add the desired amount to the function. In this case, g(x) = 5^(-x+2) + 2. Therefore, there is a vertical shift up 2 units, and this transformation applies.
6. Vertical shift down 2 units:
To shift a function vertically down, we subtract the desired amount from the function. In this case, g(x) = 5^(-x+2) - 2. Therefore, there is a vertical shift down 2 units, and this transformation applies.
In summary, the transformations that produce the graph of g(x) = 5^(-x+2) from the graph of the parent function f(x) = 5^x are:
- Reflection over the x-axis
- Horizontal shift to the left 2 units
- Horizontal shift to the right 2 units
- Vertical shift up 2 units
- Vertical shift down 2 units